The Cuntz semigroup of a C\(^*\) -algebra was introduced by J. Cuntz in 1978 with a view to examining rank functions on C\(^*\) -algebras [1]}. It has enjoyed renewed relevance in recent years owing both to its role in the classification theory of simple separable amenable C\(^*\) -algebras and to its capacity for reflecting C\(^*\) -algebraic properties in a purely algebraic setting. A salient example is the recent progress on the Toms-Winter conjecture and its confirmation in both the unique trace and stable rank one cases. This has been a concerted effort by many authors and an exhaustive list of credits would be prohibitively long, but some important works include [2]}, [3]}, [4]}, and [5]}. It has firmly established the breadth and importance of C\(^*\) -algebras which absorb the Jiang-Su algebra \(\mathcal {Z}\)  tensorially, a class that includes the approximately divisible C\(^*\) -algebras considered here ([6]}, [7]}).
